Near automorphisms of $G_{(n,m)}$

Abstract: Let $G$ be a graph with vertex set $V(G)$, $f$ a permutation of $V(G)$. Define $\delta_f(G)=|d(x,y)-d(f(x),f(y))|$ and $\delta_f(G)=\Sigma\delta_f(x,y)$, where the sum is taken over all unordered pair $x$, $y$ of distinct vertices of $G$. $\delta_f(x,U)=\Sigma\delta_f(x,y)$, where $U\subseteq V(G)$ and $y\in U$. Let $\pi(G)$ denote the smallest positive value of $\delta_f(G)$ among all permutations of $V(G)$. A permutation $f$ with $\delta_f(G)=\pi(G)$ is called a near automorphisms of $G$\cite{HV}. In this paper, we define $G_{(n,m)}$ is a graph obtained from $K_n$ by add $t_i$ pendent vertices to $y_i$ which is a vertex of $K_n$, $i=1,\cdots,m$, and we say $y_i$ is a c-pendent vertex of $G_(n,m)$. We determine $\pi(G_{(n,m)})$ and describe permutations $f$ of $G_{(n,m)}$ for which $\pi(G_{(n,m)})=\delta_f(G_{(n,m)})$. Because $G_{(1,1)}$ is a star and it is easy, hence we let $n\geq 2$. Suppose $G_(n,m)$ has $m$ c-pendent vertices ${y_1, \ldots, y_m}$ and $y_i$ has $t_i$ pendent vertices($1\leq t_1\leq t_2\leq \ldots \leq t_m$). For $m<n$ we have $$\pi(G_{(n,m)})= \left{ \begin{array}{lc} 2n-4 & n \leq t_1+2, m=1 \cr 2t_1&otherwise \end{array} \right. $$ For $m=n$ we have $$\pi(G_{(n,n)})= \left{ \begin{array}{lc} 4 & t_1=1,t_2=2 \cr 2t_1+2t_2&otherwise \end{array} \right.$$